Integrand size = 20, antiderivative size = 20 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\text {Int}\left (\frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x},x\right ) \]
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Not integrable
Time = 0.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \\ \end{align*}
Not integrable
Time = 0.58 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.10 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx \]
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Not integrable
Time = 0.17 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.90
\[\int \frac {\operatorname {arccoth}\left (1+i d +d \cot \left (b x +a \right )\right )}{x}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.80 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]
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Not integrable
Time = 1.00 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.85 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\operatorname {acoth}{\left (d \cot {\left (a + b x \right )} + i d + 1 \right )}}{x}\, dx \]
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Not integrable
Time = 5.13 (sec) , antiderivative size = 141, normalized size of antiderivative = 7.05 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]
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Not integrable
Time = 0.47 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.00 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int { \frac {\operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{x} \,d x } \]
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Not integrable
Time = 5.26 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.05 \[ \int \frac {\coth ^{-1}(1+i d+d \cot (a+b x))}{x} \, dx=\int \frac {\mathrm {acoth}\left (d\,\mathrm {cot}\left (a+b\,x\right )+1+d\,1{}\mathrm {i}\right )}{x} \,d x \]
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